DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly

DNA Tile Self-Assembly

Steffen Kopecki

Department of Computer Science

Natural Computing, Winter Term 2013/2014

S. Kopecki DNA Tile Self-Assembly 1 / 23

Table of Contents

(I) Self-Assembly Systems with a Temperature

(II) Directed vs. Undirected Self-Assembly Systems

(III) Staged Self-Assembly

(IV) Assembly of Patterns

(V) Assembly of “Smart Tiles” and “Smart Structures”

Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)

The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of

I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .

A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.

q qqp pp

q qqq qq

qqqq qq

p ppppp

An assembly is terminal if no further tiles can be attached.

A tile can attach to the growing structure if its binding strength is atleast the temperature τ .

Let τ = 2.

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

q qqp pp

q qqq qq

qqqq qq

p ppppp

Modeling of Chemical Properties

Glues are implemented by complementary DNA sticky ends u and u∗.The glue strength is the energy needed to break the hydrogen bondsbetween the sticky ends.

I the length of the sticky ends,I G,C-content (G−C pairs 3 hydrogen

bonds whereas A−T pairs have 2),I possible mismatches in u and u∗.

Depending on the temperature of the solution “weak bonds” willfrequently assemble and disassemble, but will not be stable.

Other factors can influence the glue strength, like solvents (oftensalts) in the solution.

Self-Assembly of a Counter at Temperature τ = 2

Frameqqq

q qqq qq

Half-adder0 1

sum a⊕ binput ainput bcarry a∧ b

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Frameqqq

q qqq qq

Half-adder0 1

g(qqq) = 2

g(0) = g(1) = 1

q qqq qq

Self-Assembly of DNA Sierpinski TrianglesRothemund, Papadakis, Winfree (2004)

The fractal structure of the Sierpinski triangle can also be generatedby the the xor logic gate from a string · · ·000010000 · · ·

Self-Assembly of Sierpinski Triangles

output a⊕ boutput a⊕ binput binput a

qqqqqq

0 0qqq

Self-Assembly of Sierpinski Triangles

output a⊕ boutput a⊕ binput binput a

qqqqqq

0 0qqq

Table of Contents

Directed Self-assembly Systems

An aTAM is directed (a. k. a. deterministic) if it forms one uniqueterminal assembly, where an assembly is defined by which tile type isplaced at each position.

An aTAM strictly self-assembles a shape if all of its terminalassemblies are guaranteed to have that shape, although some of theassemblies may have different tile types at the same position.

Directed Self-assembly Systems

An aTAM is directed (a. k. a. deterministic) if it forms one uniqueterminal assembly, where an assembly is defined by which tile type isplaced at each position.

An aTAM strictly self-assembles a shape if all of its terminalassemblies are guaranteed to have that shape, although some of theassemblies may have different tile types at the same position.

The Power of Undirected Systems

Theorem

For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.

Theorem

There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.

Theorem

Universality of Directed aTAM with τ = 2

Theorem

The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q

ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp

pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p

→qqq

pp p pp p pp p pp p pp p pp p pp p pp p

ppp ppp ppp ppp ppp ppp ppp ppp

pp p pp p pp p pp p pp p pp p pp p pp p ←

Open Problem

Is the directed aTAM Turing-universal at temperature τ = 1?

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

→pp p pp p pp p pp p pp p pp p pp p pp p ←

Open Problem

Theorem

· · ·��001010111�� · · ·

· · ·��101010111�� · · ·

δ(s,0) = (p,1,R)

· · ·��101010111�� · · ·

δ(p,0) = (q,0,L)

· · ·��0011101�� · · ·

I s0 0 1 0 1 0 1 1 1 H

H p1 0 1 0 1 0 1 1 1 J

I 1 p0 1 0 1 0 1 1 1 H

H q1 0 1 0 1 0 1 1 1 J

I q1 1 1 0 1 0 1 1 1 H

I 0 0 1 1 f 1 0 1 H

I 0 0 1 1 1 0 1 J

→qqq

→pp p pp p pp p pp p pp p pp p pp p pp p ←

Open Problem

Table of Contents

Staged Self-assembly

More external control is added to the assembly process by usingdifferent sets of tile types in each of several stages.

Start with a seed structure σ and sets of tile types T1, . . . ,Tn. For eachstage i = 1, . . . ,n

1. add the tile types Ti to the solution,

2. wait for a terminal structure to assemble,

3. then, “wash away” all unbound tile types.

After the n-th stage start over with the 1-st stage again.

In biochemistry wet-labs the repeated process of mixing DNAstructures (in our case tile types) into a solution and then purifying thesolution to obtain certain structures is a commonly used technique.

Staged Self-assembly

More external control is added to the assembly process by usingdifferent sets of tile types in each of several stages.

Start with a seed structure σ and sets of tile types T1, . . . ,Tn. For eachstage i = 1, . . . ,n

1. add the tile types Ti to the solution,

2. wait for a terminal structure to assemble,

3. then, “wash away” all unbound tile types.

After the n-th stage start over with the 1-st stage again.

In biochemistry wet-labs the repeated process of mixing DNAstructures (in our case tile types) into a solution and then purifying thesolution to obtain certain structures is a commonly used technique.

Universality of Staged aTAM with τ = 1

Theorem

The directed, staged aTAM at temperature τ = 1 is Turing-universal.

Staged aTAM with τ = 2 can simulate zig-zag aTAM with τ = 2.

East-west glues are actual glues while north-south glues aresimulated by the geometry of the tile.

c d ed efg

xb h ih ji k lj

Theorem

c d ed efg

xb h ih ji k lj

Theorem

c d ed efg

b h ih ji k lj

Theorem

c d ed efg

b h1x a

b h ih ji k lj

Table of Contents

What are Nanoscopic Patterns?

“Molecular pegboards” (addressable nanoarrays), which are cheap toproduce, for

a.) arranging nanoparticles (gold, silver, . . . ),

b.) molecular and logic circuits (in vitro and in vivo),

c.) enzyme interaction or enzyme detection,

d.) nano-factories like “artificial leafs”,

e.) quantum dot assembly.

a.) d.)

Pattern Assembly

grid where everynode has a property

pattern where everypixel has a color

Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Pattern Assembly

+++ � F +++ �

+ ++�

+++ F F

Minimal Tile Sets for Patterns

For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size

Theorem

A minimal tile set which strictly self-assembles a pattern P is directed.

Theorem

It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .

NP-hard: no algorithm is known which solves the problem efficiently.

Theorem

Table of Contents

Signals and Logic Gates on Tiles

Signal Passing

Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.

Logic Gates

Several signals on one tile can be combinedvia logic gates.

Signals and logic gates can be implementedusing strand displacement.

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing

Logic Gates

Signal Passing for Tile Self-AssemblyPadilla, Liu, Seeman (2011)

Smart tiles which can interac-tively self-assemble larger struc-tures are in turn self-assembledfrom smaller “DNA structures”.

Signal Passing for Tile Self-AssemblyPadilla, Liu, Seeman (2011)

Smart tiles which can interac-tively self-assemble larger struc-tures are in turn self-assembledfrom smaller “DNA structures”.

Robot Pebbles a. k. a. “Smart Sand”Gilpin, Rus et al. (2009–2012)

http://www.youtube.com/watch?v=swxTTlHjN5Q

Logic-Gated Nanorobot for Targeted TransportDouglas, Bachelet, Church (2012)

DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly

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